Visualizing Leibniz Equivalence Through Map Projections

Leibniz Equivalence tells us that different spreadings of metrical
and matter fields on the spacetime manifold represent the same physical
spacetime. That notion can be somewhat hard to grasp if one is not
comfortable with the abstract ideas of metric fields and manifolds.

It turns out, however that most of us are already familiar with
something that is pretty close to Leibniz equivalence. That is the
notion of different map projections of the earth's surface. We are all
familiar with the principal problem. The earth is round, but our maps
are flat sheets of paper. So any attempt to draw a map of the earth on
a flat sheet of paper will end up distorting the geography of the
earth.

One of the most familiar map projections of the earth's surface is
the Mercator projection:

Its problems are well-known. It renders the sizes of the land masses
well enough near the equator. The further we get from the equator, the
more it exaggerates. So both Greenland in the far north and Antarctica
in the far south are are very much bigger in the map than they are in
relation to the other continents in reality.

We learn pretty quickly how to read these projections. The key idea
is not to take distances on the page seriously. An inch on the map near
the equator corresponds to a very different distance on the earth's
surface than an inch on the map somewhere near the poles.

To start the analogy to spacetime theories:

The paper is analogous to the spacetime manifold:
the paper just provides points that will be used to describe the earth.
We don't take the distance between the points seriously.

The grid of inked lines printed on the paper is analogous
to the metric field. Those inked lines tell us what the real
distances are between different points on the map. If we know how to
read them, we quickly see that Greenland is not really as big as
Africa.

Different map projections are possible and cartographers have been
trying to devise ones that will render more correctly the different
areas of the land masses. Here are two more.

The first is a Cylindrical Equal-area projection:

The second is an Umayev cylindrical projection:

To complete the analogy, these different projections correspond to
different spreadings of the metric field over the spacetime manifold.
All three projections agree in their invariant properties; that is,
they agree in real spatial distances measured. All of them tell us the
correct size of Greenland and that Greenland is smaller than Africa.
However that same invariant information is represented by spreading the
same grid of inked lines differently over the page.

My thanks to Michel Janssen, who, through a fortuitious
misunderstanding, suggested this analogy without realizing it.

The SEP would like to congratulate the National Endowment for the Humanities on its 50th anniversary and express our indebtedness for the five generous grants it awarded our project from 1997 to 2007.
Readers who have benefited from the SEP are encouraged to examine the NEH’s anniversary page and, if inspired to do so, send a testimonial to neh50@neh.gov.